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A rainbow is an optical and meteorological phenomenon that is caused by reflection of light in water droplets. Thus, a rainbow is not an object, and cannot be physically approached.

Water drop refraction (Built with Processing and Processing.js)
Sun rays as they are refracted and reflected within the water drop. MouseY changes the drop Y position. MouseX (towards the right edge) adds more rays. The difference in the color refraction angles is amplified, for better visibility.

Geometry and Precision

As all worldly phenomena that fall into the category of sublime beauty, the physics of rainbows is hard to grasp.

The transformation of white light into multiple colors through refraction and reflection inside a water droplet - yes, that is easy. And it happens constantly, whenever light waves cross over the boundary from one medium to another. Yet sun rays enter the droplet at all possible surface points and as each ray refracts differently, colored rays go off in many different directions (within a 84 degree section). Basically there are colored rays all over, whenever there is sunlight and rain.

The magic of the rainbow happens when some of those colored rays get bundled into the eye of the observer. The geometry requires sun, rain and observer to be in perfect placement to each other. The sun behind and the rain in front of the observer - then a rainbow can be seen between the angles of 40.89 (blue) to 42 (red) degrees from the direction opposite the Sun.

Somehow the precision of nature - the precision of water droplets being perfect little spheres, and the precision of colored light's different refracting angles - enables us to experience this optical phenomenon.

What buffles me is that there are all these rainbows out there, that exist whenever sunlight hits rain, and that they are hidden in our world unless an observer positions himself at that perfect location and angle. (tbc)

Circular rainbows can be experienced from airplanes

Brocken Spectre

Why not start here?

The Scientific Explanation of Rainbows
Great antiquated video on youtube, explaining the history and physics of the rainbow. Lots of helpful illustrations / animations.

About Rainbows
By The National Center for Atmospheric Research

Topic: physics of Rainbow
Java applet showing the refraction inside a water droplet

Build a Rainbow
another applet

Physically-based Simultion of Rainbows
Physically-based model for simulating rainbows, paper presented at ACM 2012

How to make your own rainbow
Create your own rainbow effect with a glass-bead display

How to Make a Rainbow
There are 6 ways: water glass method, mirror method, CD method, ...

Rainbow Demonstration
More physic class experiments

Q: If you suddenly replaced all the water drops in a rainbow with same-sized spheres of polished diamond, what would happen to the rainbow? How do you calculate the size of a rainbow?
From the blog "Ask a Mathematician / Ask a Physicist", and because i love thought experiments.

Catching the Light - The Entwined History of Light and Mind by Arthur Zajonc

A beautiful book i can't recommend more. Chapter 7 is called “Door of the Rainbow” and talks about the scientific history of the rainbow in a poetic, philosophical and scientific way.

Beauty by Olafur Eliasson (1993)
Fine curtain of water droplets, and a suspended spotlight.

1+1=3 by Marc Quinn (2002)
White light, water, pump system

Brocken Spectre by Charles Monkhouse (2011)
Glass-bead display, as part of exhibition 'Seeing the Light, an investigation into Brocken Spectres and Heiligenschein'

Traveling Rainbows by Chris Fraser (2013)
Tiny glass beads and point light source. More optical experiments on his website.

Round Rainbow by Olafur Eliasson (2005)
Light shining onto what seems to be a prism ring.